Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

In this paper, we deal with the problem of exponential and weak stabilization for a class of distributed bilinear systems with time delay in a Hilbert space by using bounded feedback control. In the case of exponential stabilization, an explicit decay rate estimate of the stabilized state is given provided that a non-standard observability inequality condition is satisfied. The compactness hypothesis of the control operator prevents the validity of the non-standard observability inequality, thus it can be relaxed to a weaker sufficient condition to show the weak stabilization result by employing the same feedback control. Finally, numerical examples and simulations to hyperbolic and parabolic partial functional differential equations are considered to illustrate the effectiveness of the obtained theoretical results.